Method and system for locally controlling power delivery along a distribution feeder of an electricity grid

ABSTRACT

A distribution feeder of an electricity grid comprises a substation and a plurality of nodes with at least one controllable reactive power resource. A method is provided for locally controlling delivery of electrical power along the distribution feeder, wherein for a feeder segment in the distribution feeder the method comprises: obtaining an actual voltage magnitude at an upstream node and at a downstream node of the feeder segment, and a real power value at the upstream node; setting a target voltage phasor at the downstream node as a value when a power flow across the feeder segment is maintained, and when equal reactive power is injected at the upstream and downstream nodes that consumes all the reactive power in the feeder segment; and adjusting operation of the at least one controllable reactive power resource so that the actual voltage magnitude at the downstream node moves towards a target voltage magnitude of the target voltage phasor.

CROSS REFERENCE TO RELATED APPLICATION(S)

This application is a continuation of U.S. application Ser. No.16/620,195, filed Dec. 6, 2019, which is a U.S. National StageApplication of PCT/CA2018/050670, filed Jun. 5, 2018, which claims thebenefit of priority to U.S. Provisional Application No. 62/517,044,filed Jun. 8, 2017, each titled Method and System for LocallyControlling Power Delivery Along a Distribution Feeder of an ElectricityGrid, the disclosures of which are hereby incorporated by reference intheir entireties. To the extent appropriate, a claim priority is made toeach of the above disclosed applications.

FIELD

This disclosure relates generally to a method and system for locallycontrolling power delivery along a distribution feeder of an electricitygrid.

BACKGROUND

The design of electrical power distribution infrastructure (“electricitygrid”) has evolved over decades to ensure that the electrical power thatcustomers receive meet certain quality standards relating to voltage,frequency, and reliability.

In recent years, an increased penetration of solar power and otherintermittent power generation sources in the electricity grid arecausing significant control problems. For example, connected solarcapacity as low as 10% of peak capacity on a distribution feeder mayresult in voltage violations that are beyond ANSI-defined limits. Thisintermittent generation capacity must be balanced with either load orgeneration adjustments elsewhere on the electricity grid in order tomaintain system frequency. Often, a generation facility used forbalancing is located a significant distance from a feeder containing theintermittent generation source thus resulting in significant marginalpower losses, which in some cases may exceed 30%.

Also, intermittent power generation tends to cause voltage changes thatcan result in poor customer power quality and excess wear on substationtap changers. These substation tap changers incur increased maintenanceneeds and failure rates resulting from increased use caused by theintermittency. To avoid conflict between utility voltage managementsystems and voltage regulation capability on solar inverters, as well asto avoid potential poor regulation caused by customer equipment,intermittent generator operators have been forbidden from regulating thesystem voltage (IEEE 1547 and California Rule 21). Instead, electricalutilities have been monitoring line voltages and installing some in linecapability to manage voltage where needed. This approach tends to beslow in response time, and costly for the utility to implement.

Conventional electrical distribution systems are not designed toaccommodate the increasing amount of intermittent generation, and newsolutions are sought to address these challenges.

There are optimal power flow (OPF) algorithms generally known in the artdirected to minimizing loss or cost in an electrical distributionsystem. The prior art suggests a large number of optimization algorithmsand relaxations which consider constraints such as generation limits,transmission thermal limits, bus voltage limits, number of switchingoperations etc. These algorithms tend to seek to solve the followingnon-linear power flow equations:P _(k) =V _(k)Σ_(n=1) ^(N) Y _(kn) V _(n) cos(δ_(k)−δ_(n)−θ_(kn))  (1)Q _(k) =V _(k)Σ_(n=1) ^(N) Y _(kn) V _(n) sin(δ_(k)−δ_(n)−θ_(kn))  (2)where P_(k) and Q_(k) are real power (P) and reactive power (Q)delivered to bus k in a N bus system defined by Y_(kn) (Ybus matrix ofthe system) and V_(k), δ_(k) is the voltage magnitude and phase at bus kand θ_(kn) is the angle of the admittance Y_(kn).

Known OPF approaches lead to complex optimization problems requiringhigh computational resources, which can result in relatively slowreaction by power control systems executing these algorithms.

As it is desirable to respond quickly to intermittent power generationin an electrical distribution system, it is desirable to provide a meansfor controlling power delivery in an electrical distribution system thatimproves on prior art approaches.

SUMMARY

According to one aspect, there is provided a method for locallycontrolling delivery of electrical power along a distribution feeder ofan electricity grid, wherein the distribution feeder comprises asubstation and a plurality of nodes, and the plurality of nodescomprises at least one controllable reactive power resource. For afeeder segment in the distribution feeder, the method comprises:obtaining an actual voltage magnitude at an upstream node and at adownstream node of the feeder segment, and a real power value at theupstream node; setting a target voltage phasor at the downstream node asa value when a power flow across the feeder segment is maintained, i.e.equal to the real power value at the upstream node, and when totalreactive power injected at the upstream and downstream nodescollectively generates all reactive power consumed by the feedersegment; and adjusting operation of the at least one controllablereactive power resource so that the actual voltage magnitude at thedownstream node moves towards a target voltage magnitude of the targetvoltage phasor. Adjusting the operation of the at least one controllablereactive power resource can comprise using a reactive power device thatincreases reactive power to increase the actual voltage magnitude andusing a reactive power device that decreases reactive power to decreasethe actual voltage magnitude.

The actual voltage magnitude at the upstream and downstream nodes can beobtained by measuring at the feeder segment.

The reactive power that is injected at each of the upstream anddownstream nodes can be equal.

The plurality of nodes for each feeder segment can comprise at least onecontrollable real power resource, in which case the method furthercomprises: measuring an actual voltage phasor value at the upstream anddownstream nodes of the feeder segment, wherein the actual voltagephasor at the upstream and downstream nodes includes an actual voltagemagnitude and an actual phase angle relative to the substation; and,adjusting operation of the at least one controllable real power resourceso that the actual phase angle at the downstream node moves towards thetarget phase angle of the target voltage phasor.

The plurality of nodes can include a third node having an intermittentpower generation source, in which case the method further comprisesadjusting the target phasor setting at each node after a change in powergeneration from the intermittent power generation source.

According to another aspect, there is provided a system for locallycontrolling delivery of electrical power along a distribution feeder ofan electricity grid. The distribution feeder comprises a substation anda plurality of nodes. A pair of adjacent nodes define a feeder segmentof the distribution feeder and the feeder segment comprises at least onecontrollable reactive power resource. The system comprises: at least onereactive power resource controller communicative with and programmed tocontrol operation of the at least one reactive power resource; and aserver computer communicative with the at least one reactive powerresource controller. The server computer comprises a processor and amemory having encoded thereon program code executable by the processorto:

-   -   (i) receive an actual voltage magnitude at an upstream node and        at a downstream node of the feeder segment, and a real power        value at the upstream node;    -   (ii) set a target voltage phasor at the downstream node as a        value when a power flow across the feeder segment is equal to        the real power value at the upstream node, and when total        reactive power injected at the upstream and downstream nodes        collectively generates all reactive power consumed by the feeder        segment; and    -   (iii) transmit the target voltage phasor to the at least one        reactor power resource controller, such that the at least one        reactive power resource controller operates the at least one        controllable reactive power resource so that the actual voltage        magnitude at the downstream node moves towards a target voltage        magnitude of the target voltage phasor.

The system can further comprise at least one synchrophasor for measuringthe actual phasor value at each node and which is communicative with theserver computer to transmit a measured actual phasor value to the servercomputer.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a block diagram of an apparatus for providing distributedcontrol to resources on a distribution feeder of an electricity gridaccording to one embodiment.

FIG. 2 is a schematic illustration of a feeder segment and two adjacentnodes of the distribution feeder.

FIG. 3 is a flowchart illustrating execution of a distributed powerdelivery control program on the central server computer to generatetarget phasor instructions for each controlled node on the distributionfeeder according to a first embodiment.

FIG. 4 is a flowchart illustrating execution of a distributed powerdelivery control program on the central server computer to generatetarget phasor instructions for each controlled node on the distributionfeeder according to a second embodiment.

DETAILED DESCRIPTION

Embodiments described herein relate generally to a method and system forlocally controlling delivery of electrical power along a distributionfeeder in an electricity grid (herein referred to as a “local powercontrol method and system”) in a manner that reduces power loss at oneor more feeder segments of the distribution feeder. The distributionfeeder comprises a substation and a plurality of nodes, wherein thesubstation comprises controllable voltage management devices such as tapchangers (herein referred to as “utility voltage management devices”)and at least some of the nodes comprise controllable reactive powerresources, and optionally controllable real power resources. The systemincludes a server having a processor and a memory having encoded thereona distributed power delivery control program, and controllerscommunicative with the server over a network. A controller is installedat each node having a controllable reactive power resource and at thesubstation; the system also includes a controller at each of the nodeshaving a controllable real power resource when such are present. Eachcontroller is operable to control operation of the reactive and realpower resources at the nodes and the utility voltage management devicesat the substation. The control program when executed will set a targetvoltage phasor value at each node that comprises a target voltagemagnitude and a target phase angle relative to the substation. Thetarget voltage phasor at each node is selected such that the systemdelivers a target power flow along the distribution feeder, wherein thetarget power flow is based on reducing the line loss in each feedersegment by avoiding reactive power flow Q through each feeder segment,and maintaining equal real power flow P through each feeder segment.More particularly, the target power flow can be defined by twoconstraints, namely (1) the reactive power flow injected at an upstreamnode of a feeder segment and at a downstream node of the feeder segmentcollectively generates all reactive power consumed by the feedersegment, and (2) the total power flow through each feeder segment isonly the real power flow. In some embodiments, the first constraint canbe further defined to require the reactive power injected at theupstream and downstream nodes to be equal. The power delivery controlprogram instructs the controllers installed at the reactive powerresource nodes to control the operation of the reactive power resourcesso that the actual voltage magnitude of each node moves towards itstarget voltage magnitude. When real power resources are present, thepower delivery control program can instruct the controllers installed atthe real power resources nodes to control the operation of the realpower resources so that the actual phase angle of each node movestowards its target phase angle. The actual phasor value at the upstreamand downstream nodes of the feeder segment can be monitored by asynchrophaser to obtain actual voltage magnitude and actual phase angle(relative to the substation) values.

By controlling voltage levels locally using controllable real andreactive power resources, it is expected that feeder power losses willbe reduced as the need for remotely located facilities to providebalancing is reduced, and voltage levels along the feeders is expectedto fall within acceptable ranges (e.g. within ANSI defined limits) asvoltage levels are firmed along the feeder. This can be particularlyuseful on feeders that contain an intermittent generation source such asa solar power plant, which can cause frequent voltage changes.Furthermore, by basing the target power flow on the two aforementionedconstraints, a closed form solution can be provided for execution by thepower delivery control program, which should be more computationallytractable than the complex optimization problems used in known OPFapproaches, and thus enable the system to respond relatively quickly tochanges in voltage levels along the distribution feeder.

System

According to one embodiment and referring to FIG. 1 , a local powercontrol system 10 for providing local control of power delivery along adistribution feeder 11 comprises a central server computer 12,controllers 13 for controlling real power resources 15, controllers 14for controlling reactive power resources 16 along the feeder(respectively referred to as “real power resource controllers” 13 and“reactive power resource controllers” 14), and controllers 18 forcontrolling utility voltage management devices 23. The controllers 13,14, 18 are communicative with the server computer 12 over a network 19such as the Internet, either directly or with the addition of securitytunnelling hardware or software; alternatively, the server computer 12can be fitted with dedicated communication links to the controllers 13,14, 18 such as Frame Relay.

The distribution feeder 11 comprises a plurality of node sites 17(“nodes”), wherein some nodes 17 have one or more controllable reactivepower resources 16, some nodes 17 have one or more controllable realpower resources 15 and other nodes 17 have one or more non-controllableresources such as an intermittent power generation source 22. For thesake of simplicity, FIG. 1 illustrates a first node 17 having onecontrollable real power resource 15, a second node 17 having twocontrollable reactive power resources 16, namely a reactive powerconsuming device and a reactive power generating device, and a thirdnode having a non-controllable solar power generation resource 22. Thedistribution feeder also comprises a substation comprising one or moretap changers 23 and/or other utility voltage management devices.

The real and reactive power resources 15, 16 are typically located alongthe node sites 17 at locations remote from the server computer 12. Thereal power resources 15 can be electrical generators having capacity togenerate power (“generation resource”), electricity-powered deviceshaving capacity to consume a load (“load resource”), and storage deviceshaving capacity to store energy (“storage resource”) for short periodsand later release it back to the grid. Reactive power resources 16 that“generate” reactive power include capacitors, STATCOMs, solar (PV)inverters, and reactive power resources 16 that “consume” reactive powerinclude solar (PV) inverters.

In this embodiment, the controllable real power resources 15 are allload resources, and in particular comprise multiple single-speed waterpumps, analog electrical boilers, and analog electrical blowers. Thesereal power-consuming load resources 15 are normally intended to serve aprimary process other than providing local power control to a feeder(herein referred to as “process load resources”), and the power controlsystem 10 is configured to operate these load resources 15 to providelocal power control only within the operational constraints defined bythe original primary processes of these process load resources 15. Forexample, the water pumps are used primarily to regulate the water levelin a municipal water supply tank, each electrical boiler is usedprimarily to provide heat and domestic hot water for a building as partof a hybrid electric-gas heating system, and the blowers are usedprimarily to aerate a waste water treatment tank.

A load resource controller 13 is installed at the node site 17 of theprocess load resource 15 and communicates with the remotely-locatedserver computer 12. As will be explained in detail below, each loadresource controller 13 receives target phasor set-points from the servercomputer 12 comprising a target voltage magnitude and a target phaseangle, and is programmed to operate the process load resource 15 at aload set-point that causes the actual phase angle at the node site 17 tobe moved towards the target phase angle, but is also programmed to onlyoperate the process load resource 15 when the load set-point is withinthe operational constraints of the process load resource 15 (typicallydefined by the load resource's own control system). In other words, theload resource controller 13 is programmed to allow the load resource'scontrol system to override the load resource controller 13 when theoperators of the primary process require the process load resource 15 tobe used for its primary processes. For example, a municipal water plantoperator may require that a water tank be kept between 10% and 90% fullof water, and the load resource controller 13 is programmed to allow theserver computer 12 to operate the pumps for this tank while the waterlevel is within this range in order to provide local power control tothe feeder 11. However, when the water level in the tank rises to 90%full, the load resource's control system will be allowed to turn thepumps on, even if the server computer 12 desires the pumps to be keptoff. Controllable process load resources 15 which are being used at agiven time to serve their primary process are considered to be“off-line” to the server computer 12 and not available to provide localpower control; conversely, controllable load resources 15 which arewithin their primary operational constraints are considered “on-line”and available to be used to provide local power control. “Off-line” loadresources 15 are compensated for by the server computer 12 with other“on-line” load resources 15 so that the overall power controlfunctionality is preserved.

The load resource controller 13 in this embodiment is a small ruggedcomputer with capability to connect to the Internet 19, and to connectto the load resources 15 at their respective resource node sites 17. Theconnection between the load resource controller 13 and the servercomputer 12 is achieved through the internet 19, using a secure means ofcommunications. The load resource controller 13 is connected to thegeneration resource, load resource, or storage resource using one of anumber of methods, including: direct wiring to controllers or governorsof the load resource control system; direct connection to theSupervisory Control and Data Acquisition (SCADA) System used to controlthe process load resource 15 at the resource node site 17, or connectionto the network 19 used by the control system at the node site 17 thatcontrols the load resource 15. The real power resource controller 13 isalso connected to metering devices (not shown) that measure, to revenuestandards, the power being delivered or consumed by the process loadresources 15.

The load resource controller 13 may be connected to additionalmeasurement equipment (not shown) as required to ensure that operatingconstraints can be properly met, by: direct wiring to controllers ormeasurement equipment; direct connection to the SCADA System used tomeasure the process load resources 15 at the resource node site 17; orconnection to a network 19 used by the load resource's control system atthe node site 17 to measure the process load resource 15.

In operation, the load resource controller 13 will receive a targetphasor signal from the server computer 12, directing a change inconsumption or generation from one or more of the process load resources15 at the node site 17. The real power resource controller 13 willvalidate the received signal against the operating constraints of theprocess load resource 15 and clamp the signal if required. The controlsystem of the load resource 15 will send the set-point signal to theprocess load resource 15 identified by the server computer 12,commanding the requested change.

At every update interval (e.g. 2 seconds), the load resources controller13 will send a series of signals to the server computer 12,specifically:

-   -   The status or level of operation of each process load resource        15 at the resource node site 17 (there may be multiple load        resources connected to each load resource control system). The        load resource controller 13 will aggregate and send a total        power signal to the server computer 12, reflecting the power        generated or consumed at that site;    -   The load resource controller 13 will send a separate signal to        the server computer 12 to define the maximum and minimum power        levels that are available for the existing process load        resources 15 at the resource node site 17;    -   Any additional state information required by the server computer        12 to execute its costing subroutine, as will be described        below; and    -   An indicator if the load resource controller 13 itself, or the        SCADA, or the load resource control system, has suspended server        computer 12 control, and the current local control set-point if        the server computer 12 control is suspended.

The load resource controller 13 will then store the command status andthe power levels measured for every resource at the resource node site17. Data storage at the local load resource controller 13 should besufficient to maintain all records for an extended period of time, forexample two years. The server computer 12 and the load resource 15 aretime-synchronized so that all time-stamped communications between nodes17 can be properly interpreted. The control and status protocol betweenthe server computer 12 and the load resources 15 insures that networkissues (e.g. packet loss or reordering), does not cause incorrectcontrol actions. The system will run continuously, with an intendedcycle time between the server computer 12 and the load resourcecontroller 13 of about 5-10 seconds, and 5-60 seconds for largersystems. Local storage of data is maintained, time stamped in therevenue grade meters, in the server computer 12 and in the controlsystem of the load resources 15.

Like the load resource controllers 13, the reactive power resourcecontrollers 14 are located at each node site 17 of reactive powerresources 16, and are operable to control the operation of thosereactive power resources 16. The reactive power resource controller 14has the same hardware design as the load resource controller, and isprogrammed to control the reactive power resources 16. Similarly, theutility resource controller 18 is of the same hardware design as theload and reactive power resource controllers 13, 14 with programmingadapted to control the utility voltage management devices 23.

When the distribution feeder comprises both controllable real andreactive power resources, synchrophasers 20 (otherwise known as phasormeasurement units (PMU)) are installed at each node 17 and measure theactual voltage magnitude |V| and the actual voltage angle δ at everylocation that is monitored. The synchrophasers 20 are communicative withthe server computer 12 via the network 19.

The server computer 12 is a redundant server computer system, equippedwith a reliable operating system such as Linux, real time software, anda long-term database. The server computer 12 is desirably installed at asecure location, protected from unauthorized physical access, wherethere is a reliable connection to the internet and a backed-up supply ofelectricity. The server computer 12 may be a system that is spreadacross multiple hardware chassis either to aggregate sufficientprocessing capability, or to provide redundancy in the event of failure,or both. One chassis can operate as the primary server computer 12, andanother as a backup server computer 12. Each chassis can run amulti-core capable operating system.

According to alternative embodiment (not shown), the local power controlsystem 10 comprises controllers 14 for controlling reactive powerresources but does not comprise controllers for controlling real powerresources. As will be explained in detail below, each reactive powerresource controller 14 receives target voltage phasor set-points fromthe server computer 12, and is programmed to operate the reactive powerresource 16 at a set-point that causes the actual voltage magnitude atthe node site 17 to be moved towards a target voltage magnitude setpoint provided by the target voltage phasor. Since this alternativeembodiment does not involve controlling real power resources, the phaseangle along the distribution line is not controlled, and it is thus notnecessary to use synchrophasors to measure the actual phase angle ateach node. Instead, any voltage measuring means such as voltmeters (notshown) can be used to measure the actual voltage magnitude at each node.Notably, by controlling the reactive power, it is possible to influencethe phase angle difference in a system that has an X/R ratio of lineimpedance of around 1.

Power Delivery Control Program

The server computer 12 has a processor and a memory on which is stored apower delivery control program which when executed by the processorcontrols the utility voltage management devices 23 and at least thereactive power resources 16 to deliver a required amount of reactivepower to the feeder 11 to bring the voltage to acceptable levels, whileminimizing the power loss. The server computer 12 drives the real powerresource controllers 13, which then controls real power resources 15(“direct local control”). Additionally, the power delivery controlprogram can be configured to minimize the operation of the utilityvoltage management devices 23 and in particular, the substation tapchanger.

The power delivery control program uses OPF algorithms that are based ona closed form solution for radial distribution systems. Such a closedform solution is expected to be more computationally tractable thancomplex generic algorithms which tend to be relatively computationallydemanding; as a result, the power delivery control program is expectedto be able to react more quickly to changes in the distribution feederthan a program executing complex optimization problems, which isdesirable for distribution feeders containing intermittent generationsources such as solar and wind power generators.

An embodiment of the closed form solution used by the power deliverycontrol program to set the target phasor for each node will now bedescribed. This embodiment assumes a balanced, radial distributionsystem that can be reduced to a single-phase system. This embodiment isexpressed in polar form, and allows for a variable ratio of reactivepower injection at each of the upstream and downstream nodes i,j.

Referring to FIG. 2 , each feeder segment extends from an upstream nodei to a downstream node j individually, and assumes that the shuntcapacitance can be neglected. The voltage phasor at sending end V_(ij),the line current phasor I_(ij) and the line admittance Y_(ij) comprisingadmittance G_(ij)+j B_(ij), are defined as follows:

$\begin{matrix}{V_{i} = {{V_{i}}{\measuredangle\delta}_{i}}} & (3) \\{V_{j} = {{V_{j}}{\measuredangle\delta}_{j}}} & (4) \\{I_{ij} = {{I_{ij}}{\measuredangle\delta}_{I_{ij}}}} & (5) \\{Y_{ij} = {{G_{ij} + {jB}_{ij}} = \frac{1}{R_{ij} + {jX}_{ij}}}} & (6)\end{matrix}$

wherein δ_(i) is the phase angle at node i, δ_(j) is the phase angle atnode j, and δ_(l) _(ij) is the phase angle of the line current in theline segment ij.

The current (l_(ij)) in the feeder segment is defined as:

$\begin{matrix}{I_{ij} = {{\left( {V_{i} - V_{j}} \right)\mspace{14mu} Y_{ij}} = \frac{P_{ij} - {jQ}_{ij}}{V_{i}^{*}}}} & (7)\end{matrix}$

where V_(i) and V_(j) are the voltage phasors at sending and receivingend and Y_(ij) is the line admittance, P_(ij) is the real power flowfrom node i to node j and Q_(ij) is the reactive power flow from node ito node j.

P_(ij) in the feeder segment is expressed in equation (8) as:P _(ij) =Re{(V _(i) −V _(j))(Y _(ij)))*V _(i)}  (8)

The power loss P_(loss ij) in the feeder segment is:

$\begin{matrix}{P_{{loss}\mspace{14mu}{ij}} = {{R_{ij}{I_{ij}}^{2}} = {R_{ij}{\frac{P_{ij} - {jQ}_{ij}}{V_{i}^{*}}}^{2}}}} & (9)\end{matrix}$

The voltage drop from node i to node j ΔV_(ij) as defined in equation(10) is:

$\begin{matrix}{{\Delta\; V_{ij}} = {{V_{i} - V_{j}} = {\frac{{R_{ij}P_{ij}} + {X_{ij}Q_{ij}}}{V_{i}^{*}} + {j\frac{{X_{ij}P_{ij}} + {R_{ij}Q_{ij}}}{V_{i}^{*}}}}}} & (10)\end{matrix}$

If P_(ij)»Q_(ij) in (8) and (9), these equations reduce to equations(11) and (12):

$\begin{matrix}{P_{{loss}\mspace{14mu}{ij}} = {R_{ij}{\frac{P_{ij}}{V_{i}^{*}}}^{2}}} & (11) \\{{\Delta\; V_{ij}} = {\frac{R_{ij}P_{ij}}{V_{i}^{*}} + {j\frac{X_{ij}P_{ij}}{V_{i}^{*}}}}} & (12)\end{matrix}$

Equations (11) and (12) suggest that reducing the reactance flow Q_(ij)from node i to node j significantly decreases the power loss P_(loss ij)and the voltage drop ΔV_(ij), for equal P_(ij). In the proposed closedform solution, the goal is to drive the voltage at the receiving endV_(j) to prevent reactive power flow Q_(ij) in the feeder segment whilemaintaining real power flow P_(ij) through the feeder segment. Theconsumed reactive power by the line (Q_(line)=|I_(ij)|² X_(ij)) will besupplied from the two adjacent nodes according to equation (13):

$\begin{matrix}{{aQ}_{ij} = {Q_{ji} = \frac{Q_{line}}{a + 1}}} & (13)\end{matrix}$

where α is the ratio between the reactive power supply from the sendingend Q_(ij) and receiving end Q_(ji) (“Q-ratio”). The two conditions leadto equations (27) where P_(ij) is the real power flow beforeoptimization that shall be maintained and therefore is a constantobtained by equation (8):

$\begin{matrix}\left\{ \begin{matrix}{{aQ}_{ij} = Q_{ji}} \\{P_{{ij}\mspace{14mu}{new}} = P_{ij}}\end{matrix} \right. & (14)\end{matrix}$

Using the equation for complex power (S=VI*), equation (15) can beformulated:

$\begin{matrix}\left\{ \begin{matrix}{{a \cdot {{Im}\left( {V_{i}I_{ij}^{*}} \right)}} = {{Im}\left( {V_{j}\left( {- I_{ij}} \right)}^{*} \right)}} \\{{{Re}\left( {V_{i}I_{i}^{*}} \right)} = P_{ij}}\end{matrix} \right. & (15)\end{matrix}$

Substituting equations (3), (4) (5) in (14) and rearranging yieldsequation (16) with unknown |V_(j)|:

$\begin{matrix}{{\frac{\left( {P + {G_{ij}\left( {\frac{{\left( {{{Vi}^{2}a} - {Vj}^{2}} \right)B_{ij}^{2}} - {{G_{ij}\left( {P - {G_{ij}{Vi}^{2}}} \right)}\left( {a + 1} \right)}}{{\left( {a - 1} \right)B_{ij}^{2}} + {\left( {a + 1} \right)G_{ij}^{2}}} - {Vi}^{2}} \right)}} \right)^{2}}{B_{ij}^{2}} - \left( \frac{{\left( {{{Vi}^{2}a} - {Vj}^{2}} \right)B_{ij}^{2}} - {{G_{ij}\left( {P - {G_{ij}{Vi}^{2}}} \right)}\left( {a + 1} \right)}}{{\left( {a - 1} \right)B_{ij}^{2}} + {\left( {a + 1} \right)G_{ij}^{2}}} \right)^{2} - {{Vi}^{2}{Vj}^{2}}} = 0} & (16)\end{matrix}$

wherein each reference to “G”, “P”, “B” in equation (16) respectivelymeans “G_(ij)”, “P_(ij)” and “B_(ij)”

From equation (16), the voltage magnitude at the receiving end |V_(j)|,can be expressed as a function of the voltage magnitude at sending end|V_(i)|, the line admittance G_(ij), B_(ij) and power flow from node ito node j P_(ij) and Q-ratio α to produce equation (17):|V _(j) |=f(|V _(i) |, G _(ij) , B _(ij) , P _(ij), α)  (17)

Similarly, the voltage phase angle at receiving end δ_(j) can beexpressed as a function of the voltage magnitude at sending andreceiving end |V_(i)|, |V_(j)|, the line impedance G_(ij), B_(ij), andphase angle at sending end δ_(i) and Q-ratio α to produce equation (18):δ_(j) f(|V _(i) |, |V _(j) |, G _(ij) , B _(ij), δ_(i), α)  (18)

Solving equation (16) leads to a closed form solution for |V_(j)| inequation (19) and δ_(j) in equation (20):

$V_{j} = \sqrt{\frac{\begin{matrix}{{B_{ij}^{2}M} - {G_{ij}^{2}M} + {B_{ij}^{4}V_{i}^{2}} + {G_{ij}^{4}V_{i}^{2}} -} \\{{B_{ij}^{2}{aM}} - {G_{ij}^{2}{aM}} + {2G_{ij}^{4}V_{i}^{2}a} + {2B_{ij}^{2}G_{ij}^{2}V_{i}^{2}} +} \\{{B_{ij}^{4}V_{i}^{2}a^{2}} + {G_{ij}^{4}V_{i}^{2}a^{2}} - {4B_{ij}^{2}G_{ij}P_{ij}} +} \\{{2B_{ij}^{2}G_{ij}^{2}V_{i}^{2}a^{2}} + {2B_{ij}^{2}G_{ij}^{2}V_{i}^{2}a}}\end{matrix}}{{2B_{ij}^{4}} + {2B_{ij}^{2}G^{2}}}}$ where$\begin{matrix}{M = \sqrt{\begin{matrix}\left( {{B_{ij}^{2}V_{i}^{2}} + {G_{ij}^{2}V_{i}^{2}} - {2B_{ij}P_{ij}} + {B_{ij}^{2}V_{i}^{2}a} + {G_{ij}^{2}V_{i}^{2}a}} \right) \\\left( {{B_{ij}^{2}V_{i}^{2}} + {G_{ij}^{2}V_{i}^{2}} + {2B_{ij}P_{ij}} + {B_{ij}^{2}V_{i}^{2}a} + {G_{ij}^{2}V_{i}^{2}a}} \right)\end{matrix}}} & (19) \\{\delta_{j} = {\delta_{j} - {\cos^{- 1}\left( \frac{{\left( {{aV}_{i}^{2} - V_{j}^{2}} \right)B_{ij}^{2}} - {{G_{ij}\left( {P - {G_{ij}V_{i}^{2}}} \right)}\left( {a + 1} \right)}}{V_{i}{V_{j}\left( {{\left( {a - 1} \right)B_{ij}^{2}} + {\left( {a + 1} \right)G_{ij}^{2}}} \right)}} \right)}}} & (20)\end{matrix}$

Equations (19) and (20) provide a target voltage phasor at the receivingend of a feeder segment that assures that there is no Q flow while thereal power flow P_(ij) is maintained with respect to a given sending endvoltage phasor V_(i).

For a radial distribution system, the OPF solution provided in equations(19), (20) at each node can be calculated starting from the substation,successively node by node downstream to the feeder end. The advantage ofthe proposed closed-form OPF is its fast computation and that it doesnot require iterative power flow algorithms. Furthermore it works forbi-directional power flow.

System Operation

As will be discussed in more detail below, the power delivery controlprogram controls the voltage magnitude along the distribution feeder bycontrolling the operation of the reactive power resources 16 and theutility voltage management devices 23 (collectively “voltage managementdevices”) and optionally controls the phase angle δ along the feeder bycontrolling the operation of the real power resources 15. Generally, thevoltage magnitude between two adjacent nodes is similar, and the phaseangle difference will generally be small. Executing the power deliverycontrol program will determine the target phasor (voltage magnitude andphase angle) at each node that is required to deliver power to thedistribution feeder at minimized feeder power loss.

The power control program can optionally include a voltage managementdevice optimization module that can preferentially select certainvoltage management devices over others, by assigning an operating costto each voltage management device. For example, the voltage managementdevice optimization module can optionally assign a relatively highoperating cost to the substation tap changer 23 compared to the reactivepower resources 16 in order to minimize the use of the tap changer 23when controlling the voltage magnitude at each node 17 along the feeder11. The power delivery control program can also optionally include aload resource management module which comprises program code fordetermining which process load resources 15 are available to providepower control, and also to select a cost-effective combination ofavailable process load resources 15 to provide this control.

FIG. 3 is a flowchart illustrating a method for locally controllingdelivery of electrical power along a distribution feeder 11 using asystem 10 that executes the power delivery control program based on anembodiment wherein equal reactive power is injected at each node toconsume all the reactive power in the feeder segment. The system 10comprises controllers 13, 14 for controlling both controllable real andreactive power resources on the distribution feeder 11.

As noted above, the power delivery control program calculates the targetvoltage phasor for the downstream node 17 in each feeder segment of thefeeder distribution line 11 in succession, starting from the knownvoltage phasor at the substation 23 and working downstream one feedersegment at a time (the phase angle at the substation is defined to be atangle zero). Thus, the first feeder segment in the distribution line 11uses the voltage phasor at the substation 23 as the input values for thereal and imaginary parts of the voltage at the upstream node v_(ire),v_(iim) (step 100)

The power delivery control program is provided with or determines theadmittance value Y_(ij) of the feeder segment (step 102). The admittancecan be determined from the known resistance and reactance of the feedersegment.

The real power at the upstream node i can be measured through itssyncrophasor 20 using equation (5) for a feeder segment from node i tonode j when line impedance is known (step 104). Alternatively, the realpower value at the upstream node i is obtained by means known in theart, e.g. a wattmeter or state estimation. As noted above, this value isinput into the power delivery control program as the power flow P_(ij)across the feeder segment.

The inputted real and imaginary parts of the voltage at the upstreamnode v_(ire), v_(iim), the determined admittance value Y_(ij) and themeasured power flow value P_(ij) are used by the power delivery controlprogram in equations (12) and (13) to solve for the real and imaginaryparts of the voltage at the downstream node v_(jre), v_(jim). The targetvoltage phasor (comprising the target voltage magnitude and phase angle)at the downstream node is then determined by solving equations (14) and(15) (step 106).

The synchrophasor 20 at the downstream node 17 of the feeder segment isread to obtain measurements of the actual phasor at the downstream node17 (step 108). The actual phasor measurements consists of the voltagemagnitude |V| and angle δ at the downstream node 17. The real and/orreactive power controllers 13, 14 at the downstream node receive theirtarget voltage phasor from the power delivery control program, andreceive the actual voltage phasor measurements from the syncrophasor 20.With these inputs, the real and/or reactive power controllers 13, 14 candetermine the difference between the actual voltage magnitude and phaseangle and the target voltage angle and phase angle.

The real and/or reactive power controllers 13, 14 then selects one ormore real and/or reactive power resources 15, 16 to control to cause theactual voltage magnitude and phase angle at the downstream node to movetowards its target voltage magnitude and phase angle, then operatestheir selected real and/or reactive power resources 15, 16 accordingly(step 110). As noted previously, the controllable reactive powerresources 16 at node sites 17 can be used to control the reactive powerinjection at each node 17 and the substation. As is well understood bythose skilled in the art, reactive power resources 16 such as capacitorsincrease reactive power and consequently increase voltage magnitude at anode 17 and can be selected when the actual voltage magnitude is lowerthan the target voltage magnitude. Conversely, reactive power resources16 that consume reactive power and consequently decrease voltagemagnitude at a node 17 can be selected when the actual voltage magnitudeis higher than the target voltage magnitude; examples of such reactivepower resources include PV inverters and static synchronous compensators(STATSCOMs), which can be operated in inductive mode to lower thevoltage as required.

The above steps are repeated for each line segment between two nodesalong the entire distribution feeder 11. For the first line segment, thereference voltage at the upstream node will be the voltage phasor at thesubstation 23. Applying steps 100 to 110 will provide a voltage targetat the downstream node (first node 17 after the substation 23). For thesecond line segment, the voltage phasor at the upstream node will be thetarget voltage phasor at the downstream node of the first line segment.This sequence is repeated for each feeder segment until the last node isreached within the distribution feeder 11.

FIG. 4 is a flowchart illustrating a method for locally controllingdelivery of electrical power along a distribution feeder 11 using asystem 10 that executes the power delivery control program based onanother embodiment of the closed-form solution, which is implemented inthe central control server within a repeated routine. The centralcontrol server 1) obtains voltage phasors V_(i) and V_(j) at theupstream and downstream nodes using the syncrophasors 20 at each node i,j (step 200); 2) determines the admittance of the feeder Y_(ij) thencalculates the actual power flow P_(Ij) through each feeder segmentusing equation (21) (step 210); 3) determines the line admittance valuesB_(ij) and G_(ij) then calculates the target voltage phasor of thedownstream node j using the closed-form equations (32, 33) (step 220);4) ensures that target phasor is within the voltage limits of 0.95 and1.05 pu (steps 230-260) and 5) dispatches target voltage phasor to thedistributed controller at the downstream node (step 270). This routineis repeated continually for each successive pair of adjacent nodes.

In some embodiments, the real and/or reactive power resources 15, 16 donot have any operational constraints, and the power delivery controlprogram should be able to control the power delivery along the feedersegment with a minimum power loss. In other embodiments, the real and/orreactive power resources can be provided with operational constraints.For example, an operational constraint can be assigned that representsthe maximum available reactive power resources. If the maximum availablereactive power resources are not sufficient to track the voltage phasortargets, a new set of target voltage phasors for the entire feeder lineshould be computed considering the reactive power constraints. Inanother example, an operational constraint can be assigned thatrepresents a maximum threshold on the line current in each segment, asthe feeders have a maximum current constraint.

Optionally, the voltage management device optimization module can beexecuted to select a cost effective combination of voltage managementdevices 16, 23 that will be used to meet the target voltage magnitude ateach node. The voltage management device optimization module determineswhich voltage management devices 16, 23 are available to achieve thetarget voltage magnitudes at each node 17, selects a cost effectivecombination of available voltage management devices 16, 23, then sendscontrol signals to controllers of those selected voltage managementdevices 16, 23 to operate those devices accordingly. The selectedcombination can be the combination that provides the lowest operatingcost, or any one of a number of combinations which have an operatingcost below a selected threshold. Because not all nodes 17 may have areactive power resource 16 that can be controlled by the system 10, itmay not be possible to achieve the target voltage magnitudes at eachnode 17, in which case, the power delivery control program selects theavailable reactive power resources 16, 23 to come as close as possibleto the target voltage magnitude.

In some embodiments, a feeder node may not have a reactive powerresource that can be controlled, or the reactive power resource 16 at anode is not sufficient to bring the actual voltage magnitude into anallowable range or the target voltage magnitude. In this situation, theapparatus 10 may select the tap changer 23 at the substation tocontribute to meeting the target voltage magnitude at the node 17.However, because frequent use of the substation tap changer 23 isgenerally undesirable, the voltage management device optimization moduleassigns a comparatively higher operating cost to using the utilityvoltage management devices 23 and a comparatively lower operating costto using the reactive power resources 16 at the node sites 17. The costfunction for each reactive device 16, 23 is assigned based on actualcost. For example, a smart inverter can react quickly with little cost,and as a result is assigned a relatively low operating cost. Conversely,resources such as transformer tap changers that have life limits basedon operations, are assigned a relatively high operating cost. Once theoperating cost is assigned to each voltage management device 16, 23, acosting subroutine is executed to determine the available voltagemanagement devices 16, 23 and their respective voltage settings.

Optionally, each real power controller 14 can include a power deliverycontrol program that executes a process load resource management moduleto select a cost effective combination of real power resources 15 thatwill be used to meet the target phase angle at each node 17. As notedpreviously, the real power resources 15 include controllable processload resources 15 that serve a primary process, and can be used by thesystem 10 to control phase angles along the feeder 11 provided that theusage does not exceed the operational constraints dictated by the loadresource's primary process. The use and selection of such process loadresources 15 to provide load is disclosed in co-owned PCT applicationpublication no. WO 2011/085477, and is hereby incorporated by reference.

The process load resource management module includes program code whichdetermines which process load resources 15 are the most cost-effectiveto operate at any given time, then selects those process load resources15 to meet the target phasor angle at each node 17 along the feeder. Inorder to determine the relative cost to operate a process load resource15 at a particular point in time, the site control module programmingincludes a costing sub-routine which attributes a cost for operatingeach process load resource 15 at a particular point in time. The costingsubroutine takes into consideration factors such as the cost that mustbe paid to the primary process operator for using the resource 15 atthat time instance. The aggregated cost is then multiplied by a riskfactor allocated to each resource 15 at that time instance; this riskfactor takes into consideration the risk that over the period of timethe resource 16 will be used to provide power delivery control, theprimary process operator will override feeder power control and use theresource 15 for its primary purpose. The costing sub-routine thenselects a cost effective combination of process load resources to beoperated; a cost effective combination can be the combination of on-lineload resources having the lowest operational cost, or any one of acombination of load resources which fall within a defined operationalcost budget.

Once the real power resources 15 and the reactive power resources 16 areselected, the system 10 transmits a control signal to the controller 13,14 at each real and reactive power resource 15, 16 that contains thetarget phasor for the node of the real and reactive power resource 15,16. The controllers 13, 14 then operate their associated real andreactive power resource 15, 16 to achieve the target phasor. That is,the load resource controller 13 will increase the load of its loadresource when the measured phase angle at the node is lower than thetarget phase angle, and decrease the load when the measured phase angleis higher than the target phase angle. The reactive power resourcecontroller 14 will engage a reactive power resource 15 to generatereactive power at a node 17 when the measured voltage magnitude at thenode is below the target voltage magnitude, and will engage a reactivepower resource 15 to consume reactive power at a node 17 when themeasured voltage magnitude at the node 17 is above the target voltagemagnitude. In this manner, the system 10 can provide localized controlof the delivery of power to each node 17 along the feeder 11, at adesirably low feeder power loss, while keeping the substation tapchanger operation at a minimum (assuming the tap changer 23 is assigneda relatively high operational cost).

Alternatively, the real power resources 15 can include generationresources, in which case, a process generation resource managementmodule is provided to select the generation resource that will be usedto meet the target phase angle at each node. Like the load resources,the generation resources can include resources which serve a primaryprocess, in which case the system only controls those generationresources that are on-line, i.e. within the operational constraints oftheir primary process. In a manner similar to selecting a cost-effectivecombination of load resources, a costing sub-routine is executed andeach available generation resource is assigned a relative operatingcost, and the most cost-effective combination of generation resources isselected to meet the target phasor angle at each node along the feeder.Once the generation resources are selected, the system 10 sends acontrol signal to each controller of the selected generation resourcethat contains the target phasor for the node of the generation resource.The controllers then operate their associated generation resource toachieve the target phasor. That is, the generation resource controllerwill increase the generation of its load resource when the measuredphase angle at the node is higher than the target phase angle, anddecrease the generation when the measured phase angle is lower than thetarget phase angle.

According to another embodiment, the system is configured to onlycontrol the voltage magnitude along the distribution feeder. In thiscase, the power deliver control program when executed only instructionsto the reactive power resource controllers to control the operation ofthe reactive power resources, such that the actual voltage magnitude ateach node is moved towards the target voltage magnitude of the targetvoltage phasor determined by the power delivery control program.

While particular embodiments have been described in the foregoing, it isto be understood that other embodiments are possible. It will be clearto any person skilled in the art that modifications of and adjustmentsto this invention, not shown, are possible as demonstrated through theexemplary embodiment.

What is claimed is:
 1. A method for locally controlling delivery ofelectrical power along a distribution feeder of an electricity grid, thedistribution feeder comprising a substation and a plurality of nodes,and one of the plurality of nodes comprising a controllable reactivepower resource, the method comprising, for a feeder segment in thedistribution feeder: obtaining an actual voltage magnitude at anupstream node and at a downstream node of the feeder segment, and a realpower value at the upstream node; setting a target voltage phasor at thedownstream node based on the actual voltage magnitude at the upstreamnode and at the downstream node, and the real power value at theupstream node; and controlling operation of the controllable reactivepower resource based on the target voltage phasor.
 2. The method ofclaim 1, further comprising causing a tap changer of the substation tooperate based on the target voltage phasor.
 3. The method of claim 2,wherein causing the tap changer of the substation to operate is inresponse to determining the operation of the controllable reactive powerresource is insufficient to achieve the target voltage phasor.
 4. Themethod of claim 1, wherein one of the plurality of nodes comprises asecond controllable reactive power resource, the method furthercomprising: determining an operating cost of the controllable reactivepower resource and the second controllable reactive power resource; andselecting the controllable reactive power resource to be controlledbased on the operating costs.
 5. The method of claim 1, wherein theplurality of nodes comprise a plurality of additional controllablereactive power resources, the method further comprising: determining acombination of controllable reactive power resources, the combination ofcontrollable reactive power resources comprising the controllablereactive power resource and one of the plurality of additionalcontrollable reactive power resources; and controlling operation of thecombination of controllable reactive power resources based on the targetvoltage phasor.
 6. The method of claim 5, further comprising determiningan operating cost for the controllable reactive power resource and theplurality of additional controllable reactive power resources, whereindetermining the combination is based on the operating costs.
 7. Themethod of claim 1, wherein one of the plurality of nodes comprises acontrollable real power resource, the method further comprisingcontrolling operation of the controllable real power resource based onthe target voltage phasor.
 8. The method of claim 1, further comprisingdetermining an admittance of the feeder segment, wherein setting thetarget voltage phasor is further based on the admittance.
 9. A systemfor locally controlling delivery of electrical power along adistribution feeder of an electricity grid, the distribution feedercomprising a substation and a plurality of nodes, wherein a pair ofadjacent nodes define a feeder segment of the distribution feeder, andthe feeder segment comprises a controllable reactive power resource, thesystem comprising: a reactive power resource controller communicativewith and programmed to control operation of the controllable reactivepower resource; a server computer communicative with the reactive powerresource controller, and comprising a processor and a memory havingencoded thereon program code executable by the processor to: receive anactual voltage magnitude at an upstream node and at a downstream node ofthe feeder segment, and a real power value at the upstream node; set atarget voltage phasor at the downstream node based on the actual voltagemagnitude at the upstream node and at the downstream node, and the realpower value at the upstream node; and transmit the target voltage phasorto the reactive power resource controller, such that the reactive powerresource controller operates the controllable reactive power resourcebased on the target voltage phasor.
 10. The system of claim 9, whereinthe program code is further executable by the processor to cause a tapchanger of the substation to operate in response to determining theoperation of the controllable reactive power resource is insufficient toachieve the target voltage phasor.
 11. The system of claim 9, whereinthe feeder segment comprises a second controllable reactive powerresource, and wherein the program code is further executable by theprocessor to: determine an operating cost of the controllable reactivepower resource and the second controllable reactive power resource; andselect the controllable reactive power resource to be controlled basedon the operating costs.
 12. The system of claim 11, wherein the systemfurther comprises a second reactive power resource controllercommunicative with and programmed to control operation of the secondcontrollable reactive power resource, and wherein the program code isfurther executable by the processor to: transmit the target voltagephasor to the second reactive power resource controller, such that thesecond reactive power resource controller operates the secondcontrollable reactive power resource based on the target voltage phasor.13. The system of claim 9, wherein the feeder segment further comprisesa controllable real power resource, and the system further comprises areal power resource controller communicative with and programmed tocontrol operation of the controllable real power resource, and whereinthe server computer is communicative with the at least one real powerresource controller and the memory is further encoded with program codeexecutable by the processor to transmit the target voltage phasor to thereal power resource controller, such that the real power resourcecontroller operates the controllable real power resource based on thetarget voltage phasor.
 14. The system of claim 13, wherein the feedersegment comprises a second controllable real power resource, and whereinthe program code is further executable by the processor to: determine anoperating cost of the controllable real power resource and the secondcontrollable real power resource; and select the controllable real powerresource to be controlled based on the operating costs.
 15. A method forlocally controlling the voltage magnitude along a distribution feeder ofan electricity grid, the distribution feeder comprising a substation anda plurality of nodes, and comprising a plurality of controllable powerresources each located at one of the plurality of nodes, the methodcomprising, for a feeder segment in the distribution feeder: obtainingan actual voltage magnitude at an upstream node and at a downstream nodeof the feeder segment, and a real power value at the upstream node;setting a target voltage phasor at the downstream node based on theactual voltage magnitude at the upstream node and at the downstreamnode, and the real power value at the upstream node; determiningoperating costs for the controllable power resources; determining acombination of the controllable power resources to use, wherein thecombination comprises one of the plurality of controllable powerresources; and controlling operation of the combination of controllablepower resources based on the target voltage phasor.
 16. The method ofclaim 15, wherein the controllable power resources are (i) reactivepower resources or (ii) a combination of reactive power resources andreal power resources.
 17. The method of claim 15, further comprising:determining a portion of the plurality of controllable power resourcesthat are available, wherein the controllable power resources areavailable when they are within operational constraints; and determiningthe combination from the portion of the plurality of controllable powerresources that are available.
 18. The method of claim 15, furthercomprising controlling operation of a tap changer of the substation. 19.The method of claim 15, wherein the operating costs are based at leastin part on a time the controllable power resources will be used.
 20. Themethod of claim 15, further comprising assigning operational constraintsto the controllable power resources, wherein determining the combinationof the controllable power resources is further based on the operationalconstraints.